Coordinate transformations

Vector Transformation in Two Dimensions

In three dimensions, the vector transformation rule is written as

$v_i^{\text{'}}=l_{ij}{v}_j$

where $l_{ij}={𝐞}_i^{\text{'}}\bullet {𝐞}_j=\cos ({𝐞}_i^{\text{'}},{𝐞}_j)$.

In two dimensions, this transformation rule is the familiar

$\begin{array}{cc}{v}_1^{\text{'}}& =v_1\cos \theta +v_2\sin \theta \\ v_2^{\text{'}}& =-v_1\sin \theta +v_2\cos \theta \\ \end{array}$

In matrix form,

$\left[\begin{array}{c}{v}_1^{\text{'}}\\ v_2^{\text{'}}\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]$

Since we are using sines, the direction of measurement of $\theta$ is required. In this case, it is measured counterclockwise.

Tensor Transformation in Two Dimensions

In three dimensions, the second-order tensor transformation rule is written as

$T_{ij}^{\text{'}}=l_{ip}{l}_{jq}{T}_{pq}$

where $l_{ij}={𝐞}_i^{\text{'}}\bullet {𝐞}_j=\cos ({𝐞}_i^{\text{'}},{𝐞}_j)$.

The Cauchy stress $𝝈$is a symmetric second-order tensor. In two dimensions, the transformation rule for stress is then written as

$\begin{array}{cc}{\sigma }_{11}^{\text{'}}& ={\sigma }_{11}{\cos }^2\theta +{\sigma }_{22}{\sin }^2\theta +2{\sigma }_{12}\sin \theta \cos \theta \\ {\sigma }_{22}^{\text{'}}& ={\sigma }_{11}{\sin }^2\theta +{\sigma }_{22}{\cos }^2\theta -2{\sigma }_{12}\sin \theta \cos \theta \\ {\sigma }_{12}^{\text{'}}& =-{\sigma }_{11}\sin \theta \cos \theta +{\sigma }_{22}\sin \theta \cos \theta +{\sigma }_{12}({\cos }^2\theta -{\sin }^2\theta )\end{array}$

In matrix form,

$\left[\begin{array}{c}{\sigma }_{11}^{\text{'}}\\ {\sigma }_{22}^{\text{'}}\\ {\sigma }_{12}^{\text{'}}\end{array}\right]=\left[\begin{array}{ccc}{\cos }^2\theta & {\sin }^2\theta & 2\sin \theta \cos \theta \\ {\sin }^2\theta & {\cos }^2\theta & -2\sin \theta \cos \theta \\ -\sin \theta \cos \theta & \sin \theta \cos \theta & {\cos }^2\theta -{\sin }^2\theta \end{array}\right]\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{12}\end{array}\right]$

Since we are using sines, the direction of measurement of $\theta$ is required. In this case, it is measured counterclockwise.

Tensor Transformation in two Dimensions, the intrinsic approach

Let construct an orthonormal basis of the second order tensor projected in the first order tensor

$E_1=e_1\otimes e_1$

$E_2=e_2\otimes e_2$

$E_3=e_3\otimes e_3$

$E_4=\frac{1}{\sqrt{2}}(e_2\otimes e_3+e_3\otimes e_2)$

$E_5=\frac{1}{\sqrt{2}}(e_3\otimes e_1+e_1\otimes e_3)$

$E_6=\frac{1}{\sqrt{2}}(e_1\otimes e_2+e_2\otimes e_1)$

The stress and strain tensors are now defined by :

$\left\{\sigma \right\}=\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ \sqrt{2}{\sigma }_{23}\\ \sqrt{2}{\sigma }_{31}\\ \sqrt{2}{\sigma }_{12}\\ \end{array}\right\}$

and

$\left\{\epsilon \right\}=\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ \sqrt{2}{\epsilon }_{23}\\ \sqrt{2}{\epsilon }_{31}\\ \sqrt{2}{\epsilon }_{12}\\ \end{array}\right\}$

Then once constructs the bound matrix in the orthonormal base $E_i\otimes E_j$

$[\hat{R}(\theta )]=\left[\begin{array}{cccccc}{R}_{11}^2& R_{12}^2& R_{13}^2& \sqrt{2}{R}_{12}{R}_{13}& \sqrt{2}{R}_{11}{R}_{13}& \sqrt{2}{R}_{11}{R}_{12}\\ R_{21}^2& R_{22}^2& R_{23}^2& \sqrt{2}{R}_{22}{R}_{23}& \sqrt{2}{R}_{21}{R}_{23}& \sqrt{2}{R}_{22}{R}_{21}\\ R_{31}^2& R_{32}^2& R_{33}^2& \sqrt{2}{R}_{33}{R}_{32}& \sqrt{2}{R}_{33}{R}_{31}& \sqrt{2}{R}_{31}{R}_{32}\\ \sqrt{2}{R}_{21}{R}_{31}& \sqrt{2}{R}_{22}{R}_{32}& \sqrt{2}{R}_{23}{R}_{33}& R_{22}{R}_{33}+R_{23}{R}_{32}& R_{21}{R}_{33}+R_{31}{R}_{23}& R_{21}{R}_{32}+R_{31}{R}_{22}\\ \sqrt{2}{R}_{11}{R}_{31}& \sqrt{2}{R}_{12}{R}_{32}& \sqrt{2}{R}_{13}{R}_{33}& R_{12}{R}_{33}+R_{32}{R}_{13}& R_{11}{R}_{33}+R_{13}{R}_{31}& R_{11}{R}_{32}+R_{31}{R}_{12}\\ \sqrt{2}{R}_{11}{R}_{21}& \sqrt{2}{R}_{12}{R}_{22}& \sqrt{2}{R}_{13}{R}_{23}& R_{12}{R}_{23}+R_{22}{R}_{13}& R_{11}{R}_{23}+R_{21}{R}_{13}& R_{11}{R}_{22}+R_{21}{R}_{12}\end{array}\right]$

with

$[R(\theta )]$ the rotation matrix in $e_i\otimes e_j$ base.

Example

$[R(\theta )]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & \sin \theta \\ 0& -\sin \theta & \cos \theta \end{array}\right]$

is the rotation along the axis $e_1$ in the :$e_i\otimes e_j$ base

The associated rotation in the $E_i\otimes E_j$ base is :

$[\hat{R}(\theta )]=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& {\cos }^2\theta & {\sin }^2\theta & \sqrt{2}\sin \theta \cos \theta & 0& 0\\ 0& {\sin }^2\theta & {\cos }^2\theta & -\sqrt{2}\sin \theta \cos \theta & 0& 0\\ 0& -\sqrt{2}\sin \theta \cos \theta & \sqrt{2}\sin \theta \cos \theta & {\cos }^2\theta -{\sin }^2\theta & 0& 0\\ 0& 0& 0& 0& \cos \theta & -\sin \theta \\ 0& 0& 0& 0& \sin \theta & \cos \theta \end{array}\right]$

The rotation of a second order tensor is now defined by :

$\{\sigma (\theta )\}={[\hat{R}(\theta )]}^T\left\{\sigma \right\}$

Four order tensor

The élasticity tensor $C_{ijkl}$ in the :$e_i\otimes e_j\otimes e_k\otimes e_l$ is defined in the  :$E_i\otimes E_j$ by

$\left[\bar{C}\right]=\left[\begin{array}{cccccc}{C}_{1111}& C_{1122}& C_{1133}& \sqrt{2}{C}_{1123}& \sqrt{2}{C}_{1131}& \sqrt{2}{C}_{1112}\\ C_{1122}& C_{2222}& C_{2233}& \sqrt{2}{C}_{2223}& \sqrt{2}{C}_{2231}& \sqrt{2}{C}_{2212}\\ C_{1133}& C_{2233}& C_{3333}& \sqrt{2}{C}_{3323}& \sqrt{2}{C}_{3331}& \sqrt{2}{C}_{3312}\\ \sqrt{2}{C}_{1123}& \sqrt{2}{C}_{2223}& \sqrt{2}{C}_{2333}& 2C_{2323}& 2C_{2331}& 2C_{2312}\\ \sqrt{2}{C}_{1131}& \sqrt{2}{C}_{2231}& \sqrt{2}{C}_{3331}& 2C_{2331}& 2C_{3131}& 2C_{3112}\\ \sqrt{2}{C}_{1112}& \sqrt{2}{C}_{2212}& \sqrt{2}{C}_{3312}& 2C_{2312}& 2C_{3112}& 2C_{1212}\end{array}\right]$

and is rotated by:

${[\bar{C}(\theta )]}_g={[\hat{R}(\theta )]}^T\left[\bar{C}\right][\hat{R}(\theta )]$

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